__Day 10 Equations of
Circles, Circumference, and Area__

You can write an equation of a circle in a coordinate
plane if you know its radius and the coordinates of its center. Suppose the radius of a circle is ** r** and the center is (

If we square both sides, we get the standard
equation for our circle with radius *r*
and center (*h, k*).

__Standard Equation of a
Circle: (x h) ^{2}
+ (y k)^{2} = r^{2}__

If the center is the origin, then the standard
equation is *x ^{2} + y^{2}
= r^{2}.*

__GRAPHING CIRCLES__

If you know the equation of a circle, you can graph
the circle by identifying its center and radius. Heres an example. We have an equation for a circle being (*x* + 2)^{2} + (*y* 3)^{2} = 9. We can rewrite this to find our center and
radius.

(*x* (-2)^{2})
+ (*y* 3)^{2} = 3^{2}:
So our center is (-2, 3) and radius = 3.
Using this, our graph is:

__Circumference of Circles__

The __circumference__ of a circle is the distance
around the circle (just as the perimeter of a polygon is the distance around
the polygon). The ratio of the
circumference *C* of a circle to its
diameter *d* is a number slightly
greater than 3, regardless of the size of the circle. The ratio, by definition, of the
circumference of a circle to its diameter is the irrational number called __pi__
or . We have this
theorem:

If a circle has a
circumference of *C* units and a radius
of *r* units, then *C* = 2*r* or *C=**d*

The exact circumference would be a measurement of
pi. When asked to find the
circumference, usually you are looking for an estimate of the circumference.

__Area of a Circle__

When we want to measure the amount of space occupied
by a circle, we are looking for the area.
As the number of sides of a polygon inscribed in a circle increases, the
area of the polygon approached the value of *r*^{2}.

If a circle has an area of *A* square units and a radius of *r* units, then *A =**r*^{2}* *

A __sector __of a circle is the region bounded by
a central angle (or two radii) and its corresponding arc. If we want to find just the area of a sector
of the circle, we have a special formula.

If a sector of a circle has
an area of *A* square units, a central
angle measurement of *N* degrees, and a
radius of *r* units, then
.